Three Questions from Math National Contest

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Christopher Mele, By Counting His Chicks, Texas Teenager Tops Pecking Order in Math Context. New York Times, May 16, 2017.
https://www.nytimes.com/2017/05/ ... al-competition.html

Quote:

(a) "A 13-year-old boy ['who is home schooled'] from Texas won a national math competition on Monday [May 15] with an answer rooted in probabilities * * * The [white; surprisingly not Asian!] boy, Luke Robitaille, took less than a second to buzz in at the Raytheon Mathcounts National Competition [the math counter to  Scripps National Spelling Bee] with the correct answer.  The question: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?   

"As national champion, he [Luke] will receive a $20,000 college scholarship and a trip to Space Camp in Huntsville, Ala.

(b) "Contestants can use only a pencil and paper and have 45 seconds to solve word problems such as this one answered by the winner in 2014: The smallest integer of a set of consecutive integers is -32. If the sum of these integers is 67, how many integers are in the set?

(c) "(Incidentally, the winner that year [2012] answered this question correctly: A bag of coins contains only pennies, nickels and dimes with at least five of each. How many different combined values are possible if five coins are selected at random?)



Note:
(a) The last paragraph online is as follows: "(Did you give up yet? Luke’s winning response was 25 chicks. The answer to the question from last year: 67 integers. And the answer to the 2012 question about the bag of coins: 21. For more explanation, join Times readers who are discussing the answers on Facebook.)

In print, The paragraph ends at "21."
(b) I did not go to the Facebook link, because I am not a member of Facebook system.

(c) About quotation (a). My reasoning is for each quick, the chance
(i) not to be pecked is 1/4,
(ii) to be pecked once is 1/2, and
(iii) to be pecked twice (from both left and right) is 1/4.

100 chicks times 1/4 is 25.

(d) About quotation (b). I knew up to 32 will offset -32. So two integers to the right will be 33 and 34, which together make (the value of) 67. So my answer was there are 66 numbers, forgetting the integer "zero."
(e) I do not know how to approach quotation (c). It is not permutation 排列. But to me, it is not combination 組合, either.  1 1 1  5  5 has the same value as 1 1 1 1 10.  So I do not know how to approach the question.